Reply Comment: Comparison of Approaches to Classical Signature Change

TL;DR

This paper compares two approaches to classical signature change in spacetime, highlighting philosophical differences and their impact on junction conditions, without providing a rigorous derivation of distributional field equations.

Contribution

It offers an alternative postulate to Hayward's distributional approach and clarifies the influence of philosophical perspectives on the choice of junction conditions in signature change.

Findings

Difference in philosophy explains variations in results.

Darmois-Israel junction conditions are preferred over Lichnerowicz.

No rigorous derivation of distributional field equations yet.

Abstract

We contrast the two approaches to ``classical" signature change used by Hayward with the one used by us (Hellaby and Dray). There is (as yet) no rigorous derivation of appropriate distributional field equations. Hayward's distributional approach is based on a postulated modified form of the field equations. We make an alternative postulate. We point out an important difference between two possible philosophies of signature change --- ours is strictly classical, while Hayward's Lagrangian approach adopts what amounts to an imaginary proper ``time" on one side of the signature change, as is explicitly done in quantum cosmology. We also explain why we chose to use the Darmois-Israel type junction conditions, rather than the Lichnerowicz type junction conditions favoured by Hayward. We show that the difference in results is entirely explained by the difference in philosophy (imaginary versus real Euclidean ``time"), and not by the difference in approach to junction conditions (Lichnerowicz with specific coordinates versus Darmois with general coordinates).